*I can't see immediately whether or not this post constitutes as a duplicate, but I wouldn't be surprised if it does. If that is the case, then please reference me the post that I'm duplicating so that I can read through its feedback. Also, I included both the "measurements" and "measurement-problem" tags, since I'm unsure which fits better (although looking back I think it's the latter), so if you feel one or both ought to be removed, then please let me know (or do so yourself if you have the privilege to do so).
I am studying quantum mechanics, and have recently been introduced to the mathematical formalism of it. That is, that observables are interpreted as linear operators on a Hilbert space such that the observed quantities are the associated eigenvalues of the eigenfunctions (or eigenstates) of that observable. My problem has to do with the concept of the actual measurement of the observable (the act of observing, itself).
My understanding is that we say a system can be represented by a linear combination of the eigenfunctions of an operator Q (or, equivalently, that the system exists within the eigenspace of Q) if the associated eigenvalues of Q can be observed for that system. Then, upon measuring Q, the system "collapses" into a single eigenstate of Q, and then the associated eigenvalue of that state is the quantity we measure. Which state the system collapses into upon measurement is entirely probabilistic, where the associated probability is exactly the associated eigenvalue if the eigenspace is spanned, and the system is written in terms of, a normed basis.
Why does the system collapse into only one state? If we can formalize and model the system as a superposition of states when it isn't being measured, why can't we when it is? I understand that, of course, we can only measure one of the possible eigenvalues at a time, but why don't we think of that as the only eigenvalue/state we happened to measure instead of the only eigenvalue/state which exists at the moment of measurement? In other words, why do we interpret the probabilities as the probability of the system being exactly in a given state upon measurement rather than the probability that a given state just so happens to be the one we measure, even though the system is in a superposition of all possible states (at all times, not just when not being measured)?
What exactly is the precise, physical definition of a "measurement"? i.e., what set of physical phenomena constitute a measurement such that a system will collapse into a single state? When we're measuring something, we're taking something which is known and seeing how it changes due to some interaction, and considering the difference. But if the act of measurement is simply an interaction, then aren't all particles always "measuring" all other particles? That is, since in some sense, no matter how negligible, all particles are interacting with all other particles via the fundamental forces and fields, then why aren't they continually collapsing each other's superpositions into a single state? Or, in this sense, do we differentiate between an interaction and a mere "influence" (whatever that may mean), since interactions are mediated by gauge bosons and it's not like all particles are exchanging gauge bosons with all other particles at all times? In essence, what does it mean exactly for something to be measured and for something to be measuring?
I hope I've been at least somewhat clear in my phrasing of the questions. They feel quite philosophical, which may be why I had a hard time figuring out exactly what I was asking or how to say it (as opposed to a math problem which could be translated into purely logical, standard notation). If there's a way you think I could've expressed myself more clearly, don't hesitate to leave a comment (or simply make the edit yourself if it's trivial enough and you have those privileges).
Also, I've heard of the measurement problem before (but don't really know anything about it), is that what this is? Or, at the very least, is what I'm asking related to that problem in some fundamental way? I'm sure it probably is, given the nature of what I'm asking, but I've always been rather unclear on what exactly people mean when they refer to "the measurement problem".
Also, if there are any fundamental misinterpretations in the underlying concepts or formalism that I am making, then it would be greatly appreciated if you could please provide a reference so that I may read more about what exactly my mistake is. And, in general, any comments and answers are greatly appreciated. Thank you!